Trigonometric angle computer

ABSTRACT

Apparatus to calculate an angle theta of a value A cos theta , where A is known. Compare a generated signal A cos omega t with A cos theta in a differential amplifier such that when A cos omega t&gt;A cos theta there is an output pulse on an electronic time counter, the length of the pulse being directly proportional to the angle theta .

0 United States Patent in] 3,626,169

[ Inventor kouis i gl p [56] References Cited nnan a e, a. I 21 1 pp No- 8,796 UNITED STATES PATENTS [22] Filed 5 1970 3,048,337 8/1962 Byerly 235/186 [45] Patented Dem 7, 1971 3,303,493 2/1967 Charbonnier 340/347 AD [73] Assignee The United States of America 2,858,438 10/1958 Mernll 307/235 X represented by the Secretary 0! the Navy 3,054,910 9/1962 Bothwell 307/235 3,479,496 11/1969 Buesch et al 328/146 X Primary Examiner-Malcolm A. Morrison Assistant ExaminerR. Stephen Dildine, Jr. Attorneys-R. S. Sciascia, Arthur L. Branning and James G. [54] TRIGONOMETRIC ANGLE COMPUTER Murray 4 Claims, 3 Drawing Figs. v [52] U.S. Cl 235/186, ABSTRACT: Apparatus to calculate an angle 0 of a value A 307/235, 318/652, 328/146, 328/148, 340/347 SY cos 0, where A is known. Compare a generated signal A cos wt [51] Int. Cl 606g 7/22 with A cos 0 in a differential amplifier such that when A cos [50] Field of Search 235/186; wr A cos 0 there is an output pulse on an electronic time 307/231, 232, 235; 328/146-148; 340/347 SY; counter; the length of the pulse being directly proportional to 318/603, 652 the angle 0.

SWITCH TME w n 11 1-! COUNTER DIFFERENTIAL AMPLIFIER ASTAILE IULTIVIIATOR i4 IS A FILTER anon-en RECTIFIER 4H- 2 W ;i

. PAIENIEDHEE Han SHEET 1 0F 2 3.626.169

a moo "$-03 moo m Em mmwEs flmmoo $2 "3 a Am Em 22 "my mu g l d l E QR . x mEm Q3 Em INVENTOR LOU/S R. RUDOLPH BY M 01 ATTORNEY PATENTEU DEC 7 [WI sum 2 OF 2 MIC.

mmzmo G24 5 i 3 105x925? 335% N. m & HUJW H. mwESwm =2 3 5&96 I a ESE I! E "Q r- All m mom INVENTOR LOU/S R. RUDOLPH ATTORNEY TRIGONOMETRIC ANGLE COMPUTER The invention described herein may be manufactured and used by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor.

BACKGROUND OF THE INVENTION This invention relates to a system for computing the angle of a sine or cosine function. Occasions arise when an angle forming part of a trigonometric equation requires quick determination. For example, the commonly utilized equations x=A sin and FA cos 0 where .x and A are known require calculation of the angle 0. While x and A are known they do not remain constant for a long period of time and the angle 0, which is also varying, must be precisely and immediately known. Utilization of a slide rule and a trigonometric table is both cumbersome, slow and inaccurate. A digital computer must be updated with new information and is very costly.

Previous analog trigonometric calculators for calculating angles such as 0 arcsine x/A 0 arc cosine x/A were electromechanical and had the disadvantage of being too bulky, heavy and required a large power source.

SUMMARY OF THE INVENTION A technique for computing the arc sine or arc cosine is described. The method is useful when a function x=A cos 0 is given with A and x known and 0 to be determined. In general, the angle 0 is found by generating a cosine function and then making a time measurement on this waveform which is determined by the amplitude x. A basic waveform x=A cos an, is generated and when cos w! =x/A, a time I, has elapsed which defines on If 0 is known and t, measured, the angle 0 is then found, since cos 0=x/A cos an, 0 m,

A differential amplifier that compares the two waveforms, A cos 0 and A cos wt triggers a switch which output is displayed on a counter, and 1,, or 0, may thus be determined. The generated waveform may also be rectified to calculate arc sine angles.

OBJECTS OF THE INVENTION It is therefore an object of the invention to provide for an accurate system for calculation of trigonometric functions.

Another object of the present invention is to provide for a device to perform both arc sine and are cosine operations.

A still further object of this invention is to provide an analog trigonometric function calculator.

Yet another object of the present invention is to direct display of a calculated angle.

A still further object of this invention is to provide for a continuous display of a calculated angle.

Other characteristics and advantages of the invention will appear more clearly in the following description when taken in conjunction with the attached drawings in which:

provide a DESCRIPTION OF THE DRAWINGS FIG. la is a time plot of an unknown and a reference waveform.

FIG. 1b is a diagram of the waveform of the resultant output signal.

FIG. 2 is a block diagram of a circuit for the calculation of an unknown angle.

THEORY OF OPERATION Let it be assumed that for a certain period of time A sin 0 has a value. Referring to FIG. la, A sin out is generated and then rectified as indicated by the dotted lines. A sin mt rotates from i=0 to r=t,, at which time 1,, A sin mt,=.x. Since A sin 0=x and A sin wt,=x, then #0:". Further, it can be observed that the time period t, also equals time 1,. Thus, the time period t is equal to twice t, and it can be stated that wt =2 w!,=20. The angle 0 is uniquely determined by the period I,

0= (013/2 (0 being a known quantity and t; a measured quantity.

The measurement of the time period I; is more easily accomplished than the measurement of the period I, of FIG. I. It is for this reason that the electronic design of the Trigonometric Angle Computer is based on the scheme of measuring 1, as above.

The time period has a relationship to x=A cos 0 similar to that of the period 1;, to the sine function. The derivation follows the same logic.

MATHEMATICAL DERIVATION For arc sine determination Referring to FIG. I, x=A sin not (from 0 to 11 radians) r,=l/w arc sin x /A l,=l/m arc sin x lA s i+( 2) r '2 r =llw arc sin .r /A +l/w arc sin x, /A x,=x, (if A sin 0 constant for a period of A sin wt) t -2lw arc sin x,/A Let x, =x A sin 0 -;==2/w arc sin (A sin O/A) t,=2/w 6 Since w=21rf r =26/21rf"6/1rf (in radians) l =0/1rf(21r/360) (in degrees) t =0/l f By proper choice of the frequency f, e.g.,f-5555 Hz.

t (in usec.) =0 (in degrees) 7 For arc cosine determination x=A cos wt t =l/m arc cos x /A=t/21r arc cos x /A As w=21rf=21rlt [Ft/2' arc cos X A rat -1 t =t -,t/21r arc cos x /A As I; is the period t t tt/21r arc cos x; [A r.=t,+z,,r, t =r+tl2rr arc cos x /At+l/21r arc cos x /A which simplifies to t/1r arc cos x;, /A Since A cos 0=x r =!/1r arc cos A cos O/A r.=r0/-nin radians t =t6/1r (Zn/360 in degrees r,=r0/ l 80 Let t/ I 80=l Then t=l 80p.sec. (frequency 5555 Hz. (in p. sec.) 0 (in degrees) DESCRIPTION OF THE PREFERRED EMBODIMENT The operation of the angle computer is best explained by tracing the signal paths of the block diagram of FIG. 2. A known DC voltage A is fed to a chopper stage II which converts the voltage A into a square wave having a peak to peak amplitude of A. The frequency of the square wave is determined by astable multivibrator 12. The multivibrator I2, for reasons to be evident further on, has a temperature compensated frequency of 5555 Hz. The square wave chopper output is fed to a 5555 Hz. filter 13 which filters out the DC component and the harmonic frequencies. The output of the filter, having a l kHz. band-pass at 5555 Hz. is essentially sinusoidal. The gain of amplifier 14 following the filter 13 is adjustable to provide the correct amount of gain to compensate for the attenuation due to the filter I3 and thus provide an output of A sin 21: 5555 t. This is a sinusoidal waveform of amplitude A at a frequency 5555 Hz. The sinusoidal waveform is then rectified by rectifier l5 and is fed to comparator or differential amplifier 16. The other input to differential amplifier 16 is A sin which is constant for at least a period of A sin 21r 5555 I. The output of differential amplifier 16 is the difference between A sin 0 and A sin 21r 5555 r. The quiescent point of the differential amplifier 16 represents A sin 0=A sin 21r 5555 t. The differential amplifier I6 is capacitively coupled to a switch 17 which is off when A sin 6 is greater than A sin 21r 5555 tWhen A sin 0 is less than A sin 211' 5555 t, the switch 17 passes a positive pulse having a duration equal to the period that A sin 0 is less than A sin 21- 5555 t. The frequency 5555 Hz. was specifically chosen to cause the pulse width, when measured in microseconds, to equal the angle numerically. By using a time interval unit of an electronic counter 18 such as a HP 5245L/5262 A, it is possible to read the angle 0 directly in degrees on the counter readout. As explained in the analysis above, the positive pulse represents arc cos x/A, and the duration (FIG. lb) without the pulse represents arc sin x/A and it is therefore very simple to read either are sine or arc cosine off the time interval counter 18 merely by reversing the trigger slope controls. The trigonometric angle can also be read on a DC averaging-type meter, the average DC level proportional to the positive pulse width calibrated to read 1p, amp per degree of angle.

It should be obvious to those skilled in the art that the apparatus described is capable of determining arc cosine functions without utilizing rectifier 15. The astable multivibrator l2 frequency need not be 5555 Hz. and may be of any constant value that can assist in quickly determining the angle 0 from the pulse width as displayed on the counter 18.

While I have illustrated and described the best form of preferred embodiments of my invention, it will be apparent to those skilled in the art that changes may be made in the form of the apparatus without departing from the spirit of the invention as set forth in the appended claims.

l. Apparatus for calculating an unknown angle 0 comprisa first signal source having the waveform A sin 6;

a second signal source having the waveform A sin an,

wherein m! is known;

a comparator having input terminals to which said first signal source and said second signal source are directly connected and an output terminal;

means connected to said output terminal for providing a pulse wavefonn, the duration of each pulse in said waveform being proportional to said unknown angle 6;

means connected to said providing means for measuring said pulse duration and displaying said duration whereby said unknown angle 0 may be determined; and

wherein said second signal source includes:

a known DC source A;

a chopper connected to said DC source for producing a square wave output having an amplitude A;

an astable multivibrator connected to said chopper for controlling its chopping rate at a frequency m/21r;

means connected to the output of said chopper for providing a sinusoidal output whereby its output of A sin an is fed to said comparator as said second signal source.

2. Apparatus as recited in claim 3 wherein:

said pulse providing means is a switch which emits a pulse whenever A sin w! is greater than A sin 0; and wherein said measuring means is an electronic time interval counter.

3. Apparatus as recited in claim 2, including:

amplifying means and rectifying means between said sinusoidal providing means and said comparator; and wherein said sinusoidal providing means is a filter having a band-pass filter of 1 kHz. at said frequency (0/211.

4. Apparatus as recited in claim 3, wherein said multivibrator frequency is 555 5 Hz. whereby said unknown angle may be directly read off said electronic time interval counter. 

2. Apparatus as recited in claim 3 wherein: said pulse providing means is a switch which emits a pulse whenever A sin omega t is greater than A sin theta ; and wherein said measuring means is an electronic time interval counter.
 3. Apparatus as recited in claim 2, including: amplifying means and rectifying means between said sinusoidal providing means and said comparator; and wherein said sinusoidal providing means is a filter having a band-pass filter of 1 kHz. at said frequency omega /2 pi .
 4. Apparatus as recited in claim 3, wherein said multivibrator frequency is 5555 Hz. whereby said unknown angle may be directly read off said electronic time interval counter. 